Ph.D., Faculty of Science, University of Witwatersrand, 2011 / Operators in N = 4 super Yang-Mills theory with an R-charge of O(N2) are dual to backgrounds which are asymtotically AdS5 S5. In this thesis we develop e cient techniques that allow the computation of correlation functions in these backgrounds. We nd that (i) contractions between elds in the string words and elds in the operator creating the background are the eld theory accounting of the new geometry, (ii) correlation functions of probes in these backgrounds are given by the free eld theory contractions but with rescaled propagators and (iii) in these backgrounds there are no open string excitations with their special end point interactions; we have only closed string excitations. Furthermore, these correlation functions are not well approximated by the planar limit. The non-planar diagrams, which in the bulk spacetime correspond to string loop corrections, are enhanced by huge combinatorial factors. We show how these loop corrections can be resummed. As a typical example of our results, in the half-BPS background of M maximal giant gravitons we nd the usual 1=N expansion is replaced by a 1=(M +N) expansion. Further, we nd that there is a simple exact relationship between amplitudes computed in the trivial background and amplitudes computed in the background of M maximal giant gravitons. We also nd strong evidence for the BMN-type sectors suggested in arXiv:0801.4457. The problem of computing the anomalous dimensions of (nearly) half-BPS operators with a large R-charge is reduced to the problem of diagonalizing a Cuntz oscillator chain. Due to the large dimension of the operators we consider, non-planar corrections must be summed to correctly construct the Cuntz oscillator dynamics. These non-planar corrections do not represent quantum corrections in the dual gravitational theory, but rather, they account for the backreaction from the heavy operator whose dimension we study. Non-planar corrections accounting for quantum corrections seem to spoil integrability, in general. It is interesting to ask if non-planar corrections that account for the backreaction also spoil integrability. We nd a limit in which our Cuntz chain continues to admit extra conserved charges suggesting that integrability might survive.

No description available.

This thesis is concerned with the Schrödinger representation of quantum field theory. We describe techniques for solving the Schrödinger equation which supplement the standard techniques of field theory. Our aim is to develop these to the point where they can readily be used to address problems of current interest. To this end, we study realistic models such as gauge theories coupled to dynamical fermions. For maximal generality we consider particles of all physical spins, in various dimensions, and eventually, curved spacetimes. We begin by considering Gaussian fields, and proceed to a detailed study of the Schwinger model, which is, amongst other things, a useful model for (3+1) dimensional gauge theory. One of the most important developments of recent years is a conjecture by Mal-dacena which relates supergravity and string/M-theory on anti-de-Sitter spacetimes to conformal field theories on their boundaries. This correspondence has a natural interpretation in the Schrödinger representation, so we solve the Schrödinger equation for fields of arbitrary spin in anti-de-Sitter spacetimes, and use this to investigate the conjectured correspondence. Our main result is to calculate the Weyl anomalies arising from supergravity fields, which, summed over the supermultiplets of type JIB supergravity compactified on AdS(_s) x S(^5) correctly matches the anomaly calculated in the conjecturally dual N = 4 SU{N) super-Yang-Mills theory. This is one of the few existing pieces of evidence for Maldacena's conjecture beyond leading order in TV.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. Johannesburg, 2016. / The AdS/CFT correspondence is a conjectured exact duality between type IIB string theory on the AdS5 S5 background and N = 4 Super Yang-Mills theory, a conformal eld theory (CFT), on the boundary of the AdS space. A speci c observable of the CFT, which can be read from the two point correlation function, is the anomalous dimension. In this dissertation we will compute spectra of anomalous dimensions of excited giant gravitons up to two loops in a speci c limit. We are interested in the anomalous dimensions because the AdS/CFT correspondence associates them with energies of states in quantum gravity. We study operators constructed using n Z elds and m Y elds with n << m: In this case m n is a small parameter. At the leading order in m n and at large N, the problem of determining the anomalous dimensions can be mapped into the dynamics of m non-interacting magnons. The subleading terms at two loops, computed for the rst time in this dissertation, induce interactions between the magnons. Even after including this new correction, we nd the BPS operators remain BPS. / MT2016

Thesis (Ph. D.)--University of Washington, 2007. / Vita. Includes bibliographical references (p. 72-77).

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20173号 / 理博第4258号 / 新制||理||1612(附属図書館) / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)教授 大西 明, 教授 國廣 悌二, 教授 川合 光 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

This thesis is concerned with solitons (solutions of certain nonlinear partial differential equations) in certain cases when the underlying space is either a lattice or curved. Chapter 2 of the thesis is concerned with the outcome of collisions between a kink (a 1-dimensional soliton) and an antikink for certain topological discrete (TD) systems. The systems considered are the TD sine-Gordon and the TD ø(^4) For the TD sine-Gordon system it is found that the kink can support an internal shape mode which plays an important role during the collisions. In particular, this mode can be excited during collisions and this leads to spectacular resonance effects. The outcome of any particular collision has sensitive dependence on the initial conditions and could be either a trapped kink-antikink state, a "reflection" or a "transmission”. Such resonance effects are already known to exist for the conventional discrete ø(^4) system, and the TD ø(^4) system is no different, though the results for the two are not entirely similar. Chapter 3 considers the question of the existence of explicit travelling kink solutions for lattice systems. In particular, an expression for such a solution for the integrable lattice sine-Gordon system is derived. In Chapter 4, by reducing the Yang-Mills equations on the (2 + 2)-dimensional ultrahyperbolic space-time, an integrable Yang-Mills-Higgs system on (2 + 1) dimensional de Sitter space-time is derived. It represents the curved space-time version of the Bogomolny equations for monopoles on R(^3) . Using twister methods, various explicit solutions with gauge groups U(l) and SU(2) are constructed. A multi-solution SU(2) solution is also presented.

In Part I we are dealing with effective description of Yang-Mills theories based on gauge-invarint variables. For pure Yang-Mills we study the spin-charge separation varibles. The dynamics in these variables resembles the Skyrme-Faddeev model. Thus the spin-charge separation is an important intermediate step between the fundamental Yang-Mills theory and the low-energy effective models, used to model the low-energy dynamics of gluons. Similar methods may be useful for describing the Electroweak sector of the Standard Model in terms of gauge-invariant field variables called supercurrents. We study the geometric structure of spin-charge separation in 4D Euclidean space (paper III) and elaborate onconnection with gravity toy model. Such reinterpretation gives a way to see how effective flat background metric is created in toy gravity model by studying the appearance of dimension-2 condensate in the Yang-Mills (paper IV). For Electroweak theory we derive the effective gauge-invariant Lagrangian by doing the Kaluza-Klein reduction of higher-dimensional gravity with 3-brane, thus making explicit the geometric interpretation for gauge-invariant supercurrents. The analogy is then made more precise in the framework of exact supergravity solutions. Thus, we interpret the Higgs effect as spontaneous breaking of Kaluza-Klein gauge symmetry and this leads to interpretation of Higgs field as a dilaton (papers I and II). In Part II of the thesis we study rather simple field theories, called “geometric” or “instantonic”. Their defining property is exact localization on finite-dimensional spaces – the moduli spaces of instantons. These theories allow to account exactly for non-linearity of space of fields, in this respect they go beyond the standard Gaussian perturbation theory. In paper V we show how to construct a geometric theory of chiral boson by embedding it into the geometric field theory. In Paper VI we elaborate on the simplest geometric field theory – the supersymmetric Quantum Mechanics and construct new non-perturbative topological observables that have a transparent meaning both in geometric and in the Hamiltonian formalisms. In Paper VII we are motivated by making perturbations away from the simple instantonic limit. For that we need to carefully define the observables that are quadratic in momenta and develop the way to compute them in geometric framework. These correspond geometrically to bivector fields (or, in general, the polyvector fields). We investigate the local limit of polyvector fields and compare the geometric calculation with free-field approach.

Orientador: Marcos Benevenuto Jardim / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-22T22:29:57Z (GMT). No. of bitstreams: 1 Branco_LucasMagalhaesPereiraCastello_M.pdf: 1981391 bytes, checksum: 7ecd7674514f634b8bb527c0bcab1a06 (MD5) Previous issue date: 2013 / Resumo: Neste trabalho os aspectos básicos da teoria de calibre são abordados, incluindo as noções de conexão e curvatura em fibrados principais e vetoriais, considerações sobre o grupo de transformações de calibre e o espaço de moduli de soluções para a equação anti-auto-dual em dimensão quatro (o espaço de moduli de instantons). Posteriormente, mapas momento e redução são introduzidos. Primeiramente, no contexto clássico de geometria simplética e depois no contexto de geometria hyperkähler. Por fim, são apresentadas aplicações da teoria de mapas momento e redução em teoria de calibre. As equações ADHM são introduzidas e mostra se que estas podem ser dadas como o conjunto de zeros de um mapa momento hyperkähler. Além disso, considerações são feitas acerca da construção ADHM de instantons, que relaciona soluções dessas equações com as soluções da equação de anti-auto-dualidade. O espaço de moduli de conexões planas é também abordado. Neste caso, a curvatura é vista como um mapa momento e os cálculos podem ser generalizados para o espaço de moduli de conexões planas sobre variedades Kähler de dimensões mais altas e para o espaço de moduli de instantons sobre variedades hyperkähler de dimensão quatro / Abstract: In this work it is developed the basic concepts of gauge theory, including the notions of connections and curvature on principal bundles and vector bundles, considerations on the group of gauge transformations and the moduli space of anti-self-dual connections in dimension four (the instanton moduli space). After, moment maps and reduction are introduced. First in the classical context of symplectic geometry, then in hyperkähler geometry. At last, applications to the theory of moment maps and reduction in gauge theory are given. The ADHM equations are introduced and it is shown that solutions to these equations can be given by the zeros of a hyperkähler moment map. Furthermore, the ADHM construction, that relates the ADHM equations to instanton solutions, is discussed. The moduli space of flat connections over a Riemann surface is also treated. In this case, the curvature is seen as a moment map and the calculations can be generalized to flat connections over higher-dimensional Kähler manifolds and to the instanton moduli space over four dimensional hyperkähler manifolds / Mestrado / Matematica / Mestre em Matemática

L'objectif de ce travail est une construction perturbative rigoureuse de la théorie de la Yang-Mills SU(2) dans l'espace euclidien à quatre dimensions. La technique d'intégration fonctionnelle donne une basemathématique pour établir les équations de flot différentielles du groupe de renormalisation pour l'action efficace. Si l'introduction de régulateurs dans l'espace de moments permet de donner une définition mathématique des fonctions de Schwinger, la difficulté importante de l'approche est le fait que cesrégulateurs brisent l'invariance de jauge. Ainsi, le travail principal est alors de prouver à tous les ordres en perturbation l'existence de ces fonctions de correlation et la validité des identités de Slavnov-Taylor pour la théorie renormalisée. / The goal of this work is a rigorous perturbative construction of the SU(2) Yang-Mills theory in four dimensional Euclidean space. The functional integration technique gives a mathematical basis for establishing the differential Flow Equations of the renormalization group for the effective action. While the introduction of momentum space regulators permits to give a mathematical definition of the Schwinger functions, the important difficulty of the approach is the fact that these regulators break gauge invariance. Thus the main part of the work is to prove at all loop orders the existence of the vertex functions and the restoration of the Slavnov-Taylor identities in the renormalised theory.